On the Gorenstein property for modular invariants

Let G⊂GL(V) be a finite group, where V is a finite dimensional vector space over a field F of arbitrary characteristic. Let S(V) be the symmetric algebra of V and S(V)^G the ring of G-invariants. We prove here the following results:. TheoremSuppose that G contains no pseudo-reflection (of any kind).(1)If S(V)^G is Gorenstein, then G⊂SL(V).(2)If G⊂SL(V) then the Cohen-Macaulay locus of S(V)^G coincides with its Gorenstein locus. In particular if S(V)^G is Cohen-Macaulay then it is also Gorenstein.This extends well-known results of K. Watanabe in case (charF,|G|)=1. It also confirms a special case of a conjecture due to G. Kemper, E. Körding, G. Malle, B.H. Matzat, D. Vogel and G. Wiese. A similar extension is given to D. Benson's theorem about the Gorenstein property of (S(V)⊗Λ(V))^G, the polynomial tensor exterior algebra invariants. Our proof uses non-commutative algebra methods in an essential way.